120 research outputs found

    Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

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    We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments

    Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

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    Elliptic partial differential equations with diffusion coefficients of lognormal form, that is a=exp(b)a=exp(b), where bb is a Gaussian random field, are considered. We study the â„“p\ell^p summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of bb. These summability results have direct consequences on the approximation rates of best nn-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of bb, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Lo\`eve representation of bb may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion

    Single Bunch Instabilities in FCC-ee

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    FCC-ee is a high luminosity lepton collider with a centre-of-mass energy from 91 to 365 GeV. Due to the machine parameters and pipe dimensions, collective effects due to electromagnetic fields produced by the interaction of the beam with the vacuum chamber can be one of the main limitations to the machine performance. In this frame, an impedance model is required to analyze these instabilities and to find possible solutions for their mitigation. This paper will present the contributions of specific machine components to the total impedance budget and their effects on the beam stability. Single bunch instability thresholds will be estimated in both transverse and longitudinal planes

    Stable high-order randomized cubature formulae in arbitrary dimension

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    We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure μ\mu defined on a domain Γ⊆Rd\Gamma \subseteq \mathbb{R}^d, in any dimension dd. Each cubature formula is conceived to be exact on a given finite-dimensional subspace Vn⊂L2(Γ,μ)V_n\subset L^2(\Gamma,\mu) of dimension nn, and uses pointwise evaluations of the integrand function ϕ:Γ→R\phi : \Gamma \to \mathbb{R} at m>nm>n independent random points. These points are distributed according to a suitable auxiliary probability measure that depends on VnV_n. We show that, up to a logarithmic factor, a linear proportionality between mm and nn with dimension-independent constant ensures stability of the cubature formula with high probability. We also prove error estimates in probability and in expectation for any n≥1n\geq 1 and m>nm>n, thus covering both preasymptotic and asymptotic regimes. Our analysis shows that the expected cubature error decays as n/m\sqrt{n/m} times the L(Γ,μ)L(\Gamma, \mu)-best approximation error of ϕ\phi in VnV_n. On the one hand, for fixed nn and m→∞m\to \infty our cubature formula can be seen as a variance reduction technique for a Monte Carlo estimator, and can lead to enormous variance reduction for smooth integrand functions and subspaces VnV_n with spectral approximation properties. On the other hand, when we let n,m→∞n,m\to\infty, our cubature becomes of high order with spectral convergence. As a further contribution, we analyse also another cubature formula whose expected error decays as 1/m\sqrt{1/m} times the L2(Γ,μ)L^2(\Gamma,\mu)-best approximation error of ϕ\phi in VnV_n, and is therefore asymptotically optimal but with constants that can be larger in the preasymptotic regime. Finally we show that, under a more demanding (at least quadratic) proportionality betweeen mm and nn, the weights of the cubature are positive with high probability

    Numerical analysis of the factorization method for EIT with a piecewise constant uncertain background

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    International audienceWe extend the factorization method for electrical impedance tomography to the case of background featuring uncertainty. We first describe the algorithm for the known but inhomogeneous backgrounds and indicate expected accuracy from the inversion method through some numerical tests. Then we develop three methodologies to apply the factorization method to the more difficult case of a piecewise constant but uncertain background. The first one is based on a recovery of the background through an optimization scheme and is well adapted to relatively low-dimensional random variables describing the background. The second one is based on a weighted combination of the indicator functions provided by the factorization method for different realizations of the random variables describing the uncertain background. We show through numerical experiments that this procedure is well suited to the case where many realizations of the measurement operators are available. The third strategy is a variant of the previous one when measurements for the inclusion-free background are available. In this case, a single pair of measurements is sufficient to achieve comparable accuracy to the deterministic case

    Numerical analysis of the Factorization Method for Electrical Impedance Tomography in inhomogeneous medium

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    The retrieval of information on the coefficient in Electrical Impedance Tomography is a severely ill-posed problem, and often leads to inaccurate solutions. It is well known that numerical methods provide only low-resolution reconstructions. The aim of this work is to analyze the Factorization Method in the case of inhomogeneous background. We propose a numerical scheme to solve the dipole-like Neumann boundary-value problem, when the background coefficient is inhomogeneous. Several numerical tests show that the method is capable of recovering the location and the shape of the inclusions, in many cases where the diffusion coefficient is nonlinearly space-dependent. In addition, we test the numerical scheme after adding artificial noise

    Sparse polynomial approximation of parametric elliptic PDEs Part I: affine coefficients

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    International audienceWe consider the linear elliptic equation −div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ¯ a + j≥1 y j ψ j for some parameter vector y = (y j) j≥1 ∈ U = [−1, 1] N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H 1 0 (D). We consider both Taylor series and Legendre series. Previous results [8] show that, under a uniform ellipticity assuption, for any 0 p. We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. Our analysis applies to other types of linear PDEs with similar affine parametrization of the coefficients

    Coupling impedances and collective effects for FCC-ee

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    A very important issue for the Future Circular Collider (FCC) is represented by collective effects due to the selfinduced electromagnetic fields, which, acting back on the beam, could produce dangerous instabilities. In this paper we will focus our work on the FCC electron-positron machine: in particular we will study some important sources of wake fields, their coupling impedances and the impact on the beam dynamics. We will also discuss longitudinal and transverse instability thresholds, both for single bunch and multibunch, and indicate some ways to mitigate such instabilities
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